1,721,456 research outputs found
Diffusion-aggregation processes with mono-stable reaction terms
No full textThis paper analyses front propagation of the equation v_t=[D(v)v_x]_x +f(v), t≥0, x∈R, where f is a monostable (i.e. Fisher-type) nonlinear reaction term and D(v) changes its sign once, from positive to negative values, in the interval v ∈ [0, 1] where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density v at time t and position x. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states v ≡ 0 and v ≡ 1. In the degenerate case, i.e. when D(0) = 0 and/or D(1) = 0, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both
Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis
No full textCurrent interest in pattern formation can be traced to a seminal paper by Turing, who demonstrated that a system of reacting and diffusing chemicals, called morphogens, can interact so as to produce stable nonuniform concentration patterns in space. Recently, a Turing model has been suggested to explain the development of pigmentation patterns on species of growing angelfish such as Pomacanthus semicirculatus, which exhibit readily observed changes in the number, size, and orientation of colored stripes during development of juvenile and adult stages, but the model fails to predict key features of the observations on stripe formation. Here we develop a generalized Turing model incorporating cell growth and movement, we analyze the effects of these processes on patterning, and we demonstrate that the model can explain important features of pattern formation in a growing system such as Pomacanthus. The applicability of classical Turing models to biological pattern formation is limited by virtue of the sensitivity of patterns to model parameters, but here we show that the incorporation of growth results in robustly generated patterns without strict parameter control. In the model, chemotaxis in response to gradients in a morphogen distribution leads to aggregation of one type of pigment cell into a striped spatial pattern
A chemotactic model for the advance and retreat of the primitive streak in avian development
No full textThe formation of the primitive streak in early avian development marks the onset of gastrulation, during which large scale cell movement leads to a trilaminar blastoderm comprising prospective endodermal, mesodermal and ectodermal tissue. During streak formation a specialized group of cells first moves anteriorly as a coherent column, beginning from the posterior end of the prospective anterior-posterior axis (a process called progression), and then reverses course and returns to the most posterior point on the axis (a process called regression). To date little is known concerning the mechanisms controlling either progression or regression. Here we develop a model in which chemotaxis directs the cell movement and which is capable of reproducing the principal features connected with progression and regression of the primitive streak. We show that this model exhibits a number of experimentally-observed features of normal and abnormal streak development, and we propose a number of experimental tests which may serve to illuminate the mechanisms. This paper represents the first attempt to model the global features of primitive streak formation, and provides an initial stage in the development of a more biologically-realistic discrete cell model that will allow for variation of properties between cells and control over movement of individual cells. (C) 2000 Society for Mathematical Biology
Development and applications of a model for cellular response to multiple chemotactic cues
No full textThe chemotactic response of a cell population to a single chemical species has been characterized experimentally for many cell types and has been extensively studied from a theoretical standpoint. However, cells frequently have multiple receptor types and can detect and respond chemotactically to more than one chemical. How these signals are integrated within the cell is not known, and we therefore adopt a macroscopic phenomenological approach to this problem. In this paper we derive and analyze chemotactic models based on partial differential (chemotaxis) equations for cell movement in response to multiple chemotactic cues. Our derivation generalizes the approach of Othmer and Stevens [29], who have recently developed a modeling framework for studying different chemotactic responses to a single chemical species. The importance of such a generalization is illustrated by the effect of multiple chemical cues on the chemotactic sensitivity and the spatial pattern of cell densities in several examples. We demonstrate that the model can generate the complex patterns observed on the skin of certain animal species and we indicate how the chemotactic response can be viewed as a form of positional indicator
Spatial pattern formation in chemical and biological systems
Get PDFOne of the central issues in developmental biology is the formation of spatial pattern in the embryo. A number of theories have been proposed to account for this phenomenon. The most widely studied is reaction diffusion theory, which proposes that a chemical pre-pattern is first set up due to a system of reacting and diffusing chemicals, and cells respond to this pre-pattern by differentiating accordingly. Such patterns, known as Turing structures, were first identified in chemical systems only recently. This article reviews the application of reaction diffusion theory to chemical systems and then considers a number of biological applications
Pattern formation in a generalised chemotactic model
Get PDFMany models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate, numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development
Wave patterns in one-dimensional nonlinear degenerate diffusion equations
Get PDFSeveral different types of wave patterns occur in physiology, chemistry and biology. In many cases such phenomena are modelled by reactive-diffusive parabolic systems (see, for example, Fisher 1937; Kolmogorov et al. 1937; Winfree 1988; Murray 1989; Swinney & Krinsky 1992). In many biological and physical situations, dispersal is modelled by a density-dependent diffusion coefficient, for example, the bacterium Rhizobium diffuses through the roots of some leguminosae plants according to a nonlinear diffusive law (Lara-Ochoa & Bustos 1990); nonlinear diffusion has been observed in the dispersion of some insects (Okubo 1980) and small rodents (Meyers & Krebs 1974)
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